A projective structure on a smooth manifold consists of an equivalence class \(\mathfrak{p}\) of torsion-free connections on its tangent bundle, where two such connections are called equivalent if they have the same geodesics up to parametrisation.

The representative connections of a projective structure \(\mathfrak{p}\) on a smooth manifold \(M\) are in one-to-one correspondence with the sections of an affine bundle \(A \to M\), whose total space carries a split-signature metric \(h_{\mathfrak{p}}\) as well as a symplectic form \(\Omega_{\mathfrak{p}}\), both of which are defined in a canonical fashion from \(\mathfrak{p}\). Consequently, all the submanifold notions of (pseudo-)Riemannian geometry and symplectic geometry can be applied to the representative connections of \(\mathfrak{p}\). This point of view gives rise to the notion of a minimal Lagrangian connection.

The aim of this research project is to investigate various questions related to minimal Lagrangian connections, in particular, to provide a characterisation of projective manifolds that arise from a minimal Lagrangian connection.

## Publications

## Team Members

**JProf. Dr. Thomas Mettler**

Project leader

Goethe-Universität Frankfurt

mettler(at)math.uni-frankfurt.de